Question

The number of points, where the curve y = x⁵ – 20x³ + 50x + 2 crosses the x-axis, is _____.

Answer 1

Correct Answer: 5

We are given the function \( y = x^5 - 20x^3 + 50x + 2 \). To find the points where the curve crosses the x-axis, we need to solve for \( y = 0 \). \includegraphics[width=0.75\linewidth]{5.png} \[ \frac{dy}{dx} = 5x^4 - 60x^2 + 50 = 5x^4 - 12x^2 + 10 \] We solve: \[ \frac{dy}{dx} = 0 \quad \Rightarrow \quad x^4 - 12x^2 + 10 = 0 \] Solving for \( x^2 \), we find: \[ x^2 = 6 \pm \sqrt{26} \quad \Rightarrow \quad x^2 \approx 6 \pm 5.1 \] Thus: \[ x \approx \pm 3.3, \pm 0.95 \] Therefore, the number of points where the curve cuts the x-axis is 5.